Abstract
In this paper, we present the idea of interval valued fuzzy subgroup defined over a certain t-conorm (Γ-IVFSG) and prove that every IVFSG is Γ-IVFSG. We use this ideology to define the concepts of Γ-IVF cosets, Γ-IVFNSG and formulate their various important algebraic characteristics. We also propose the study of the notion of level subgroups of Γ-IVFSG and investigate the condition under which a Γ-IVFS is Γ-IVFSG. Moreover, we extend the study of this phenomenon to introduce the concept of quotient group of a group Z relative to the Γ-IVFNSG and acquire a correspondence between each Γ-IVF(N)SG of a group Z and Γ-IVF(N)SG of its quotient group. Furthermore, we define the index of Γ-IVFSG and establish the Γ-interval valued fuzzification of Lagrange's theorem of any Γ-IVFSG of a finite group Z.
Highlights
In the late eighteenth century, Lagrange’s Theorem appeared in the literature
Pietro Abbati gave the first complete proof of this theorem about thirty years after the Lagrange’s modification. This theorem has a significant role in the development of modern group theory
It is an incredible asset to investigate finite groups; as it gives an exact review about subgroups of a finite group
Summary
In the late eighteenth century, Lagrange’s Theorem appeared in the literature. It was basically discovered to resolve the problem of finding roots of the equation of degree greater than 4 and its association with symmetric volumes. The theory of fuzzy logic offers a mathematical method to apprehend the uncertainty related to human cerebral process like thoughtful and intellectual. It handles issues of uncertainty and lexical imprecision. The limitation of this theory is the case when which we do not have exact information of the membership function In these cases, it is reasonable to declare each component of the fuzzy set of membership grades by methods of interval. It is reasonable to declare each component of the fuzzy set of membership grades by methods of interval These perceptions rise the development of fuzzy sets called the theory of IVFS. We conclude this section by establishing the -interval valued fuzzification of Lagrange’s theorem of -IVFSG
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